On the Geometry of Projective Tensor Products

ON THE GEOMETRY OF PROJECTIVE TENSOR PRODUCTS

OHAD GILADI, JOSCHA PROCHNO, CARSTEN SCHUTT,¨ NICOLE TOMCZAK-JAEGERMANN, AND ELISABETH WERNER

n Abstract. In this work, we study the volume ratio of the projective tensor products `p ⊗π n n `q ⊗π `r with 1 ≤ p ≤ q ≤ r ≤ ∞. We obtain asymptotic formulas that are sharp in almost all cases. As a consequence of our estimates, these spaces allow for a nearly Euclidean decomposition of Kaˇsintype whenever 1 ≤ p ≤ q ≤ r ≤ 2 or 1 ≤ p ≤ 2 ≤ r ≤ ∞ and q = 2. Also, from the Bourgain-Milman bound on the volume ratio of Banach spaces in terms of their cotype 2 constant, we obtain information on the cotype of these 3-fold projective tensor products. Our results naturally generalize to k-fold products `n ⊗ · · · ⊗ `n with k ∈ p1 π π pk N and 1 ≤ p1 ≤ · · · ≤ pk ≤ ∞.

1. Introduction In the geometry of Banach spaces the volume ratio vr(X) of an n-dimensional normed space X is defined as the n-th root of the volume of the unit ball in X divided by the volume of its John ellipsoid. This notion plays an important role in the local theory of Banach spaces and has significant applications in approximation theory. It formally originates in the works [Sza78] and [STJ80], which were influenced by the famous paper of B. Kaˇsin[Kaˇs77] on nearly Euclidean orthogonal decompositions. Kaˇsindiscovered that for arbitrary n ∈ N, 2n the space `1 contains two orthogonal subspaces which are nearly Euclidean, meaning that n their Banach-Mazur distance to `2 is bounded by an absolute constant. S. Szarek [Sza78] n noticed that the proof of this result depends solely on the fact that `1 has a bounded volume n ratio with respect to `2 . In fact, it is essentially contained in the work of Szarek that if X is a 2n-dimensional Banach space, then there exist two n-dimensional subspaces each n having a Banach-Mazur distance to `2 bounded by a constant times the volume ratio of X squared. This observation by S. Szarek and N. Tomczak-Jaegermann was further investigated in [STJ80], where the concept of volume ratio was formally introduced, its connection to the cotype 2 constant of Banach spaces was studied, and Kaˇsintype decompositions were proved n n for some classes of Banach spaces, such as the projective tensor product spaces `p ⊗π `2 , 1 ≤ p ≤ 2. Given two vector spaces X and Y , their algebraic tensor product X ⊗ Y is the subspace of the dual space of all bilinear maps on X × Y spanned by elementary tensors x ⊗ y, x ∈ X, y ∈ Y (a formal definition is provided below). The theory of tensor products was established by A. Grothendieck in 1953 in his Résumé[Gro53] and has a huge impact on

Date: April 6, 2017. 2010 Mathematics Subject Classiﬁcation. 46A32, 46B28. O.G. was supported in part by the Australian Research Council. J.P. was supported in part by the FWF grant FWFM 162800. A part of this work was done when N.T-J. held the Canada Research Chair in Geometric Analysis; also supported by NSERC Discovery Grant. E.W. was supported by NSF grant DMS-1504701.

1 Banach space theory (see, e.g., the survey paper [Pis12]). This impact and the success of the concept of tensor products is to a large extent due to the work [LP68] of J. Lindenstrauss and A. Pe lczynskiin the late sixties who reformulated Grothendieck’s ideas in the context of operator ideals and made this theory accessible to a broader audience. Today, tensor products appear naturally in numerous applications, among others, in the entanglement of qubits in quantum computing, in quantum information theory in terms of (random) quantum channels (e.g., [AS06,ASW10,ASW11,SWZ11]) or in theoretical computer science to represent locally decodable codes [Efr09]. For an interesting and recently discovered connection between the latter and the geometry of Banach spaces we refer the reader to [BNR12]. The geometry of tensor products of Banach spaces is complicated, even if the spaces involved are of simple geometric structure. For example, the 2-fold projective tensor product of Hilbert spaces, `2 ⊗π `2, is naturally identified with the Schatten trace class√S1, the space ∗ of all compact operators T : `2 → `2 equipped with the norm kT kS1 = trace T T . This space does not have local unconditional structure [GL74]. The geometric structure of triple tensor products is even more complicated and therefore it is hardly surprising that very little is known about the geometric properties of these spaces. For instance, regarding the permanence of cotype (see below for the definition) under projective tensor products, it was proved by N. Tomczak-Jaegermann in [TJ74] that the space `2 ⊗π `2 has cotype 2, but the corresponding question in the 3-fold case is still open for more than 40 years. G. Pisier proved in [Pis90] and [Pis92] that the space Lp ⊗π Lq has cotype max (p, q) if p, q ∈ [2, ∞) and, till the present day, it is unknown whether these spaces have a non-trivial cotype when p ∈ (1, 2) and q ∈ (1, 2]. In the recent paper [BNR12] by J. Briët,A. Naor and O. Regev 1 1 1 they showed that the spaces `p ⊗π `q ⊗π `r fail to have non-trivial cotype if p + q + r ≤ 1. Their proof uses deep results from the theory of locally decodable codes. A direct, rather surprising consequence of their work is that for p ∈ (1, ∞) the space `p ⊗π `2p/(p−1) ⊗π `2p/(p−1) fails to have non-trivial cotype, while, by Pisier’s result, the 2-fold projective tensor product `2p/(p−1) ⊗π `2p/(p−1) has finite cotype. Let us also mention that interest in cotype is, to a great deal, due to a famous result of B. Maurey and G. Pisier [MP76], who showed that a n Banach space X fails to have finite cotype if and only if it contains `∞’s uniformly. In view of the various open questions and surprising results around the geometry of projective tensor products, with the present paper, we contribute to a better understanding of the n n n geometric structure of 3-fold projective tensor products `p ⊗π `q ⊗π `r (1 ≤ p ≤ q ≤ r ≤ ∞) by studying one of the key notions in local Banach space geometry, the volume ratio of these spaces. This provides new structural insight and allows, on the one hand, to draw conclusions regarding cotype properties of these spaces and, on the other hand, to see which of these spaces allow a nearly Euclidean decomposition of Kaˇsintype.

2. Presentation of the main result

Given two Banach spaces, (X, k · kX ) and (Y, k · kY ), the projective tensor product space, denoted X ⊗π Y , is the space X ⊗ Y equipped with the norm ( m m ) X X kAkX⊗πY = inf kxikX kyikY : A = xi ⊗ yi, xi ∈ X, yi ∈ Y . (2.1) i=1 i=1 See Section 3.3 below for more information on projective tensor products.

2 Given an n-dimensional Banach space (X, k · kX ), let BX denote its unit ball, and let EX be the ellipsoid of maximal volume contained in BX . The volume ratio of X is deﬁned by vol (B )1/n vr(X) = n X , (2.2) voln(EX )

where voln(·) denotes the n-dimensional Lebesgue measure. Considering the importance of tensor products, it is natural to study the geometric properties of X ⊗π Y and, in particular, the volume ratio vr(X ⊗π Y ) of these spaces. As already n n mentioned in the introduction, when X = `p (1 ≤ p ≤ 2) and Y = `2 this was carried out in [STJ80]. The complete answer was given later by C. Sch¨uttin [Sch82], where it was proved that if 1 ≤ p ≤ q ≤ ∞, then  1, q ≤ 2,  1 − 1 n 2 q , p ≤ 2 ≤ q, 1 + 1 ≥ 1, n n  p q vr `p ⊗π `q p,q 1 − 1 n p 2 , p ≤ 2 ≤ q, 1 + 1 ≤ 1,  p q  max 1 − 1 − 1 ,0 n ( 2 p q ), p ≥ 2.

The notation p,q means equivalence up to constants that depend only on p and q. In [DM05] this was generalized by A. Defant and C. Michels to the setting E ⊗π F , where E and F are symmetric Banach sequence spaces, each either 2-convex or 2-concave. n n n We study tensor products `p ⊗π `q ⊗π `r (1 ≤ p ≤ q ≤ r ≤ ∞). Our main result is as follows.

Theorem A. Let n ∈ N and 1 ≤ p ≤ q ≤ r ≤ ∞. Then we have  1, r ≤ 2,  1 1 1  max( 2 − q − r ,0) 1 1 1 n n n n , p ≤ 2 ≤ q, p + q + r ≥ 1, vr `p ⊗π `q ⊗π `r p,q,r 1 − 1 n p 2 , p ≤ 2 ≤ q, 1 + 1 + 1 ≤ 1,  p q r  max 1 − 1 − 1 − 1 ,0 n ( 2 p q r ), p ≥ 2. In the case p ≤ q ≤ 2 ≤ r, we have

1 1 1 1 n n n min( 2 − r , q − 2 ) 1 .p,q,r vr `p ⊗π `q ⊗π `r .p,q,r n .

Here and in what follows .p,q,r, &p,q,r mean inequalities with implied positive constants that depend only on the parameters p, q, r. The asymptotic notation p,q,r means that we have both .p,q,r and &p,q,r. Remark 2.1. We would like to remark that whenever 1 ≤ p ≤ q ≤ 2 ≤ r ≤ ∞, we are able to improve on the general bound given by TheoremA in the following situations (see Corollary 5.7):

 min 1 − 1 , 1 − 1 n ( q 2 2 r ), p = 1 ≤ q ≤ 2 ≤ r,  n n n 1 − 1 vr `p ⊗π `q ⊗π `r p,q,r n q 2 , p = q ≤ 2, r = ∞, 1, 1 ≤ p ≤ q = 2 ≤ r.

3 n n n Remark 2.2. TheoremA and Remark 2.1 immediately imply that the spaces `p ⊗π `q ⊗π `r allow a nearly Euclidean decomposition of Kaˇsintype, when 1 ≤ p ≤ q ≤ r ≤ 2 or 1 ≤ p ≤ 2 ≤ r ≤ ∞ and q = 2. Before we proceed, let us comment on the strategy of the proof. Recall ﬁrst that a Banach space (X, k · kX ) is said to have enough symmetries if the only operators that commute with every isometry on X are multiples of the identity. It is known that if (X, k · kX ) is n-dimensional and has enough symmetries, then EX is given by n −1 n EX = kid : `2 → Xk B2 , n where B2 denotes the n-dimensional Euclidean ball (see, for instance, [TJ89, Sec. 16]). Hence, using formula (2.2), if (X, k · kX ) is n-dimensional and has enough symmetries, then 1/n voln(BX ) n (∗) √ 1/n n vr(X) = n id : `2 → X n voln(BX ) id : `2 → X , (2.3) voln(B2 ) n 1/n √ where in (∗) we used Stirling’s formula to deduce vol(B2 ) 1/ n. It is also known that projective tensor products of `p spaces are spaces with enough symmetries [Sch82, STJ80] n n n and therefore formula (2.3) holds for the spaces `p ⊗π `q ⊗π `r . Thus, in order to prove n n n TheoremA, it is enough to compute the volume of the unit ball of `p ⊗π `q ⊗π `r , and the n3 n n n norm of the natural identity between `2 and `p ⊗π `q ⊗π `r . The rest of this paper is organized as follows. In Section3 we collect some basic results n n n which will be used later. In Section4 we estimate the volume of the unit ball in `p ⊗π `q ⊗π `r . In Section5 we estimate the norm of the natural identity. In Section6 we discuss the case of k-fold tensor products. Finally, in Section7, we present some applications of TheoremA.

3. Preliminaries In this section, we introduce the necessary notions and background material and provide the main ingredients needed to prove the estimates for the volume ratio of 3-fold projective tensor products. These include an extension of Chevet’s inequality, a lower bound on the volume of unit balls in Banach spaces due to Sch¨utt,the famous Blaschke-Santal´oinequality, and a multilinear version of an inequality of Hardy and Littlewood.

3.1. General notation. Given a Banach space (X, k · kX ), denote its unit ball by BX and its dual space by X∗. For two Banach spaces X and Y , we write L(X,Y ) for the space of all bounded linear operators from X to Y . n n For 1 ≤ p ≤ ∞, `p is the vector space R with the norm ( Pn |x |p1/p , 1 ≤ p < ∞ n i=1 i k(xi)i=1kp = max |xi| , p = ∞. 1≤i≤n n n n ∗ The unit ball in `p is denoted by Bp = {x ∈ R : kxkp ≤ 1}. The conjugate p of p is 1 1 deﬁned via the relation p + p∗ = 1. Unless otherwise stated, e1, . . . , en will be the standard unit vectors in Rn. We shall also use the asymptotic notations . and & to indicate the corresponding inequalities up to universal constant factors, and we shall denote equivalence up to universal

4 constant factors by , where A B is the same as (A . B) ∧ (A & B). If the constants involved depend on a parameter α, we denote this by .α, &α and α, respectively.

3.2. Polar body and Blaschke-Santalóinequality. A convex body K in Rn is a compact convex subset of Rn with non-empty interior. The n-dimensional Lebesgue measure of a n convex body K ⊆ R is denoted by voln(K). If 0 is an interior point of a convex body K in Rn, we define the polar body of K by ◦ n K := y ∈ R : ∀x ∈ K : hx, yi ≤ 1 , where h·, ·i stands for the standard inner product on Rn. Note that the unit ball of any norm on Rn is a convex body and its polar body is just the unit ball of the corresponding dual norm. Moreover, we have (K◦)◦ = K. The famous Blaschke-Santalóinequality provides a sharp upper bound for the volume product of a convex body with its polar (see, for example, [Pis89, Sec. 7] or [Sch14]). Lemma 3.1 (Blaschke-Santalóinequality). Let K be an origin symmetric convex body in Rn. Then ◦ n 2 voln(K) · voln(K ) ≤ voln(B2 ) , with equality if and only if K is an ellipsoid. 3.3. Tensor products and extended Chevet inequality. Given two Banach spaces (X, k · kX ) and (Y, k · kY ), the algebraic tensor product X ⊗ Y can be constructed as the space of linear functionals on the space of all bilinear forms on X × Y . Given x ∈ X, y ∈ Y and a bilinear form B on X × Y , we define (x ⊗ y)(B) := B(x, y). On the tensor product space define the projective tensor product space, denoted by X ⊗π Y , as X ⊗ Y equipped with the norm ( m m ) X X kAkX⊗πY := inf kxikX kyikY : A = xi ⊗ yi, xi ∈ X, yi ∈ Y . i=1 i=1

Also, deﬁne the injective tensor norm space, denoted X ⊗ Y , as the tensor product space X ⊗ Y equipped with the norm

( n m ) X X kAk := sup ϕ(x )ψ(y ) : A = x ⊗ y , ϕ ∈ B ∗ , ψ ∈ B ∗ . X⊗Y i i i i X Y i=1 i=1 Here, we only consider ﬁnite-dimensional spaces and therefore the tensor products are always ∗ complete. In this case, it can be shown that X ⊗π Y = N (X ,Y ), the space of all nuclear ∗ ∗ operators from X into Y . We will often use the fact that (X ⊗π Y ) , the dual space of ∗ ∗ X ⊗π Y , is the space of operators from X to Y , L(X ,Y ), equipped with the standard operator norm. It is also known that L(X∗,Y ) can be identiﬁed with the injective tensor product X ⊗ Y . In particular, for A ∈ X ⊗ Y , we have

kAkX⊗Y = sup kAxkY . x∈BX∗ We refer the reader to [Rya02] and [DFS08] for more information about tensor products.

5 n Recall that for a sequence x = (xi)i=1 in a Banach space (X, k · kX ), the norm kxkω,2 is given by 1 n ! 2 X 2 kxkw,2 := sup |ϕ(xi)| . kϕkX∗ =1 i=1 The following inequality is due to Chevet [Che78].

Lemma 3.2 (Chevet’s inequality). Let (X, k · kX ), (Y, k · kY ) be two Banach spaces and m,n m n consider sequences x1, . . . , xm ∈ X, y1, . . . , yn ∈ Y , and sequences (gi,j)i,j=1, (ξi)i=1, (ηj)j=1 of independent identically distributed standard Gaussians random variables. Then m,n n m X m X n X E gi,jxi ⊗ yj ≤ k(xi)i=1k E ηjyj + (yj)j=1 E ξixi . (3.1) ω,2 ω,2 i,j=1 X⊗Y j=1 Y i=1 X

∗ ∗ ∗ ∗ It is known that BX ⊗πY = conv (BX ⊗ BY ) (see, for example, Proposition 2.2 in [Rya02]) and thus

(xi ⊗ yj)i,j ω,2 = (xi)i ω,2 · (yj)j ω,2. (3.2) Using inequalities (3.1) and (3.2), we obtain the following 3-fold version of Chevet’s inequality.

Lemma 3.3 (3-fold Chevet inequality). Let (X, k · kX ), (Y, k · kY ), (Z, k · kZ ) be Banach spaces. Assume that x1, . . . , xm ∈ X, y1, . . . , yn ∈ Y and z1, . . . , z` ∈ Z. Let gi,j,k, ξi, ηj, ρk, i = 1, . . . , m, j = 1, . . . , n, k = 1, . . . , `, be independent standard Gaussians random variables. Then m,n,` X E gijk xi ⊗ yj ⊗ zk ≤ Λ,

i,j,k=1 X⊗Y ⊗Z where ` n m n X m ` X Λ := k(xi)i=1k (yj)j=1 E ρkzk + k(xi)i=1k (zk)k=1 E ηjyj w,2 w,2 w,2 w,2 k=1 Z j=1 Y m n ` X + (yj)j=1 (zk)k=1 E ξixi . w,2 w,2 i=1 X We would like to point out that, simply using the triangle inequality, a corresponding lower bound can be obtained up to an absolute constant. 3.4. Volume ratio and Rademacher cotype. The concept of Rademacher cotype was introduced to Banach space theory by J. Hoﬀmann-Jørgensen [HJ74] in the early 1970s. The basic theory was developed by B. Maurey and G. Pisier [MP76]. A Banach space (X, k · kX ), is said to have Rademacher cotype α if there exists a constant C ∈ (0, ∞) such that for all m ∈ N and all x1, . . . , xm ∈ X, m m 1/α X α X kxikX ≤ C E εixi . (3.3) X i=1 i=1 ∞ In (3.3) and in what follows, (εi)i=1 denotes a sequence of independent symmetric Bernoulli 1 random variables, that is, P(εi = 1) = P(εi = −1) = 2 for all i ∈ N. The smallest C

6 that satisﬁes (3.3) is denoted by Cα(X) and called the cotype α constant of X. By taking x1 = x2 = ··· = xm it follows that necessarily α ≥ 2. 1 1 1 In [BNR12] it was shown that, whenever p + q + r ≤ 1, then

Cα (`p ⊗π `q ⊗π `r) = ∞, for every 2 ≤ α < ∞. However, it is still an open question whether for example `2 ⊗π `2 ⊗π `2 has ﬁnite cotype. One reason to study volume ratio of tensor products is its relation to the cotype constant. More precisely, given an estimate on volume ratio, one can use the following result by Bourgain and Milman [BM87]) which connects volume ratio and cotype of a Banach space. Theorem 1 ([BM87]). Let X be a Banach space. Then

vr(X) . C2(X) log (2C2(X)) . The relation between volume ratio and cotype property is far from being well understood. For example, it was asked in [STJ80] whether bounded volume ratio implies cotype q for every q > 2. n n n In Section7 below we discuss the cotype property of the space `p ⊗π `q ⊗π `r , as well as the extended case of k-fold tensor products for k > 3. 3.5. Volume of unit balls in Banach spaces. The following result is a special case of Lemma 1.5 in [Sch82]. It provides a lower estimate for the volume of the unit ball of a n normed space by the volume of a B∞ ball of a certain radius, arising from an average over sign vectors.

Lemma 3.4. Let (X, k · kX ) be an n-dimensional Banach space, and let e1, . . . , en be basis ∞ vectors such that keikX = 1, and (εi)i=1 a sequence of independent symmetric Bernoulli random variables. Then n !−n n X 2 E εiei ≤ voln(BX ).

i=1 X We will use this result in combination with the 3-fold version of Chevet’s inequality to n n n obtain a lower bound on the volume of the unit ball in `p∗ ⊗ `q∗ ⊗ `r∗ which gives, using the Blaschke-Santal´oinequality, an upper bound on the volume of the unit ball in the dual space `p ⊗π `q ⊗π `r, as well as tensor products of more than three `p spaces. 3.6. Rademacher versus Gaussian averages. In order to use Chevet’s inequality in combination with Lemma 3.4, we need to pass from a Rademacher average to a Gaussian one. The following result due to Pisier shows that Rademacher averages are dominated by Gaussian averages in arbitrary Banach spaces. Note however that in general these averages are not equivalent.

Lemma 3.5 ([Pis86]). Let (X, k · kX ) be a Banach space, 1 ≤ p < ∞ and let ξ1, . . . , ξn be independent, symmetric random variables. Assume that E|ξi| = E|ξj| for all 1 ≤ i, j ≤ n. Then, for all x1, . . . , xn ∈ X, we have n p n p X −p X E εixi ≤ E|ξ1| E ξixi .

i=1 i=1

7 In particular, if we choose g1, . . . , gn to be independent Gaussian random variables, then −p p/2 since E|g1| = (π/2) , we have n p n p X X E εixi p E gixi . . i=1 i=1 3.7. A multilinear Hardy-Littlewood type inequality. An essential tool in proving n3 n n n upper bounds on the norm of the natural identity between `2 and `p ⊗π `q ⊗π `r is the following inequality, which is a generalization of a classical inequality by Hardy and Littlewood [HL34]. For now, we state it only for the case of 3-fold tensors. In Section6 we also present the consequences of the general version.

1 1 1 1 Theorem 2 ([PP81], Thm. B). Let n ∈ N and 1 ≤ p, q, r ≤ ∞ so that p + q + r ≤ 2 . Then, n n n for all A ∈ `p∗ ⊗ `q∗ ⊗ `r∗ , we have

kAk n3 kAk`n ⊗ `n ⊗ `n , `µ . p∗ q∗ r∗ where µ is given by 3 µ := 1 1 1 . (3.4) 2 − p − q − r

n n n 4. The volume of the unit ball in `p ⊗π `q ⊗π `r n n n In this section we evaluate the volume of the unit ball of `p ⊗π `q ⊗π `r up to constants depending only on the parameters p, q, r. The main ingredients in the proof are the 3-fold version of Chevet’s inequality (Lemma 3.3) and the Blaschke-Santal´oinequality (Lemma 3.1). The next theorem will be the consequence of the following two subsections.

Theorem B. Let n ∈ N and 1 ≤ p ≤ q ≤ r ≤ ∞. Then − min( 1 , 1 )−min( 1 , 1 )− 1 −1 1/n3 − min( 1 , 1 )−min( 1 , 1 )− 1 −1 p 2 q 2 r 3 n n n p 2 q 2 r n ≤ voln B`p ⊗π`q ⊗π`r .p,q,r n . 4.1. Lower bounds on the volume. Let us first we fix some more notation. For 1 ≤ p, q, r ≤ ∞ and n ∈ N, let n n n n Xπ := `p ⊗π `q ⊗π `r , where we suppress the parameters p, q, r that are clear from the context. For the correspond- n n n n ing norm, we write in short k·kπ instead of k·kXπ . Similarly, we define X := `p∗ ⊗ `q∗ ⊗ `r∗ n n and write k · k instead of k · kX . We denote the corresponding unit balls by Bπ and B n respectively. Any A ∈ X , we express in the form n X A = Ai,j,k ei ⊗ ej ⊗ ek. i,j,k=1 The main tool in proving a lower bound is the following estimate that compares the n3 injective norm k · k with the `1-norm in R .

8 n Proposition 4.1. Let n ∈ N, 1 ≤ p, q, r ≤ ∞ and assume A ∈ X . Then n 1 − min 1 , 1 −min 1 , 1 − 1 −1 X kAk ≥ n ( p 2 ) ( q 2 ) r |A | . 2 i,j,k i,j,k=1

n Proof. To prove this, we identify X with a space of operators. We have

n X kAk = max Ai,j,k xiyjzk . kxkp=kykq=kzkr=1 i,j,k=1 In what follows, ε, δ, η ∈ {−1, 1}n denote sign vectors. We divide the proof into three diﬀerent cases, depending on which side of 2 the parameters p, q, r lie. − 1 Case 1: Assume that 2 ≤ p ≤ q. In this case we choose x = n p (ε1, . . . , εn), y := − 1 − 1 n q (δ1, . . . , δn), and z = n r (η1, . . . , ηn). Then we obtain

n n X − 1 − 1 − 1 X max Ai,j,kxiyjzk ≥ n p q r max Ai,j,k εiδjηk kxk =kyk =kzk =1 ε,δ,η∈{−1,1}n p q r i,j,k=1 i,j,k=1 n n − 1 − 1 − 1 X X = n p q r max Ai,j,k δjηk δ,η∈{−1,1}n i=1 j,k=1 n n − 1 − 1 − 1 X 1 X X ≥ n p q r max Ai,j,k δjηk . η∈{−1,1}n 2n i=1 δ∈{−1,1}n j,k=1 Applying Khintchine’s inequality and then H¨older’sinequality, we obtain

n n n n n 1/2 X 1 X X 1 X X X 2 max Ai,j,k δjηk ≥ √ max Ai,j,k ηk η∈{−1,1}n 2n 2 η∈{−1,1}n i=1 δ∈{−1,1}n j,k=1 i=1 j=1 k=1 n n n 1 X X X ≥ √ max Ai,j,k ηk . η∈{−1,1}n 2n i=1 j=1 k=1 Again, applying Khinchine’s inequality and then H¨older’s inequality,

n n n n n n X X X X X 1 X X max Ai,j,k ηk ≥ Ai,j,k ηk η∈{−1,1}n 2n i=1 j=1 k=1 i=1 j=1 η∈{−1,1}n k=1

n n n !1/2 1 X X X 2 ≥ √ |Ai,j,k| 2 i=1 j=1 k=1 n 1 X ≥ √ |Ai,j,k|. 2n i,j,k=1

n Case 2: Assume that p ≤ 2 ≤ q. In this case, we choose for x ∈ Bp the standard unit vectors e1, . . . , en and y, z as in the previous case. We get, again by using the inequalities of

9 Khintchine and H¨older, n n X − 1 − 1 X max Ai,j,k xiyjzk ≥ n q r max max Ai,j,k δjηk kxk =kyk =kzk =1 1≤i≤n δ,η∈{−1,1}n p q r i,j,k=1 j,k=1 n n − 1 − 1 X X = n q r max max Ai,j,k ηk 1≤i≤n η∈{−1,1}n j=1 k=1 n n − 1 − 1 −1 X X ≥ n q r max Ai,j,k ηk η∈{−1,1}n i,j=1 k=1 n n − 1 − 1 −1 X 1 X X ≥ n q r A η 2n i,j,k k i,j=1 η∈{−1,1}n k=1 n n !1/2 1 − 1 − 1 −1 X X 2 ≥ √ n q r |Ai,j,k| 2 i,j=1 k=1 n 1 − 1 − 1 − 3 X ≥ √ n q r 2 |Ai,j,k|. 2 i,j,k=1 n n Case 3: Assume that p ≤ q ≤ 2. Choose for x ∈ Bp and y ∈ Bq the standard unit vectors e1, . . . , en and z as in the previous two cases. We have n n X − 1 X max Ai,j,k xiyjzk ≥ n r max max Ai,j,k ηk kxk =kyk =kzk =1 1≤i,j≤n η∈{−1,1}n p q r j,k=1 k=1 n − 1 X = n r max |Ai,j,k| 1≤i,j≤n k=1 n − 1 −2 X ≥ n r |Ai,j,k|. i,j,k=1 This completes the proof of the proposition. As an immediate consequence of Lemma 4.1, we obtain a lower bound on the volume n n radius of the unit ball Bπ in Xπ . Corollary 4.2. Let n ∈ N and 1 ≤ p, q, r ≤ ∞. Then we have 1/n3 1 1 1 1 1 n − min( p , 2 )−min( q , 2 )− r −1 voln3 Bπ ≥ n . Proof. From Lemma 4.1, we obtain that

1 1 1 1 1 3 n min( p , 2 )+min( q , 2 )+ r +1 n B ⊆ 2 n B1 . Switching to the polar bodies implies

n 1 − min 1 , 1 −min 1 , 1 − 1 −1 n3 B ⊇ n ( p 2 ) ( q 2 ) r B . π 2 ∞ Taking volumes and the n3-rd root, the previous inclusion immediately gives 1/n3 1 1 1 1 1 n − min( p , 2 )−min( q , 2 )− r −1 voln3 Bπ ≥ n , which completes the proof.

10 4.2. Upper bounds on the volume. To compute the matching upper bound on the vol- n ume radius of Bπ , we will use Lemma 3.4. To be more precise, the idea is as follows. Using our extended version of Chevet’s inequality (see Lemma 3.3), we obtain a lower bound for n the volume of B from Lemma 3.4. We then use the Blaschke-Santal´oinequality (see Lemma n 3.1) to derive an upper bound on the volume of Bπ . The next proposition will be a consequence of the 3-fold Chevet inequality, where we n n n n apply Lemma 3.3 to the space X and choose (xi)i=1, (yj)j=1, (zk)k=1 to be the standard basis vectors. ∞ Proposition 4.3. Let n ∈ N and 1 ≤ p ≤ q ≤ r ≤ ∞. Let (εi,j,k)i,j,k=1 be a sequence of independent Bernoulli random variables. Then we have n 1 1 1 1 1 X max ∗ , +max ∗ , + ∗ −1 E εi,j,k ei ⊗ ej ⊗ ek p,q,r n p 2 q 2 r . (4.1) . i,j,k=1 Proof. First, recall that the Rademacher average is smaller than the Gaussian average (see Lemma 3.5) and so it is enough to prove inequality (4.1) with Gaussian random variables. In order to do that, recall the well known fact that, for all 1 ≤ α < ∞, and standard Gaussian random variables g1, . . . , gn, n X 1/α E giei α n . n i=1 `α n Also, it is known that if we consider the standard unit vectors e1, . . . , en in `α, then we have

1 1 1 n max α , 2 − 2 (ei)i=1 ω,2 = n . Then, Lemma 3.3 implies n 1 1 1 1 1 X max ∗ , +max ∗ , + ∗ −1 E εi,j,kei ⊗ ej ⊗ ek p,q,r n p 2 q 2 r . i,j,k=1 max 1 , 1 +max 1 , 1 + 1 −1 max 1 , 1 +max 1 , 1 + 1 −1 + n p∗ 2 r∗ 2 q∗ + n q∗ 2 r∗ 2 p∗ max 1 , 1 +max 1 , 1 + 1 −1 ≤ 3n p∗ 2 q∗ 2 r∗ , (4.2) where in the last inequality we used the assumption that p ≤ q ≤ r. n An upper bound on the volume radius Bπ is now an immediate consequence of Lemma 3.4, Proposition 4.3, and the Blaschke-Santal´oinequality. Corollary 4.4. Let n ∈ N and 1 ≤ p ≤ q ≤ r ≤ ∞. Then we have 1/n3 1 1 1 1 1 n − min( p , 2 )−min( q , 2 )− r −1 voln3 Bπ .p,q,r n . n Proof. Applying Proposition 4.3 to X and using Lemma 3.4, we get 1/n3 1 1 1 1 1 n − max p∗ , 2 −max q∗ , 2 − r∗ +1 voln3 B &p,q,r n . (4.3) Lemma 3.1 implies that 3 3 1/n n 1/n3 n1/n n3 2 −3 voln3 (Bπ ) · voln3 B ≤ volnk B2 n . (4.4)

11 Thus, combining (4.3) and (4.4), we obtain

1/n3 1 1 1 1 1 n max p∗ , 2 +max q∗ , 2 + r∗ −4 voln3 Bπ .p,q,r n . 1 1 1 1 Since max p∗ , 2 − 1 = − min p , 2 , we have 1/n3 1 1 1 1 1 n − min( p , 2 )−min( q , 2 )− r −1 voln3 Bπ .p,q,r n , and the proof is complete.

5. The operator norm of the identity operator n3 n In this section we will present upper and lower bounds for id ∈ L(`2 ,Xπ ) that are sharp for most choices of p, q, r. The lower bounds are based on Chevet’s inequality, on the special n structure of the space of diagonal tensors in Xπ or the choice of particular 3-fold tensors. To obtain upper bounds, we use the results for 2-fold projective tensor products and a generalized version of an inequality that was proved by Hardy and Littlewood to study bilinear forms. The main theorem in this section reads as follows.

Theorem C. Let n ∈ N and 1 ≤ p ≤ q ≤ r ≤ ∞. Then we have  1 + 1 n 2 r , r ≤ 2,  1 1 1  max( q + r , 2 ) 1 1 1 n3 n n , p ≤ 2 ≤ q, p + q + r ≥ 1, kid : `2 → Xπ k p,q,r 1 + 1 + 1 − 1 n p q r 2 , p ≤ 2 ≤ q, 1 + 1 + 1 ≤ 1,  p q r  max 1 + 1 + 1 , 1 − 1 n ( p q r 2 ) 2 , 2 ≤ p. In the case q ≤ 2 ≤ r, we obtain the following bounds,

1 1 3 min 1 + 1 ,1 2 + r n n ( q r ) n .p,q,r kid : `2 → Xπ k . n .

The lower bound is proved in Section 5.1 and the upper bound in Section 5.2. Also, in Section 5.3 we show how the lower bound can be improved in some special cases when q ≤ 2 ≤ r in the previous theorem. In what follows, we will use the shorthand notation n3 n kidk n3 n for kid : `2 → Xπ k. `2 →Xπ 5.1. Lower bounds on the operator norm. The following lower bound can be obtained n by considering the space of diagonal tensors in Xπ and by using the 3-fold version of Chevet’s inequality (see Proposition 4.3).

Lemma 5.1. Let n ∈ N and 1 ≤ p, q, r ≤ ∞. Then we have 1 1 1 1 1 1 1 1 1 1 max(min(1, p + q + r )− 2 ,0) min( p , 2 )+min( q , 2 )+ r − 2 kidk n3 n p,q,r max n , n . (5.1) `2 →Xπ & n Proof. By Theorem 1.3 in [AF96], it is known that the space of diagonal tensors in Xπ is n3 isometric to `s , where s is given by 1 s := . 1 1 1 min p + q + r , 1

12 Hence, we have

Pn Pn ei ⊗ ei ⊗ ei i,j,k=1 ei ⊗ ei ⊗ ei i,j,k=1 `n3 1 1 π s s − 2 kidk`n3 →Xn ≥ = = n . 2 π Pn Pn ei ⊗ ei ⊗ ei ei ⊗ ei ⊗ ei i,j,k=1 n3 i,j,k=1 n3 `2 `2

Since kidk n3 n ≥ 1, we have in fact `2 →Xπ

1 1 1 1 max(min(1, p + q + r )− 2 ,0) kidk n3 n ≥ n . (5.2) `2 →Xπ n Next, notice that by Proposition 4.3, there exists a choice of signs (εi,j,k)i,j,k=1 such that n X max 1 , 1 +max 1 , 1 + 1 −1 ( p∗ 2 ) ( q∗ 2 ) r∗ εi,j,ke1 ⊗ e2 ⊗ e3 .p,q,r n . i,j,k=1 On the other hand, n X 3 2 εi,j,k e1 ⊗ e2 ⊗ e3 = n . i,j,k=1 n3 `2 Thus, Pn εi,j,k e1 ⊗ e2 ⊗ e3 i,j,k=1 n3 `2 kidk`n3 →Xn = kidk(Xn)∗→`n3 ≥ 2 π π 2 Pn i,j,k=1 εi,j,k e1 ⊗ e2 ⊗ e3 5 1 1 1 1 1 (∗) 1 1 1 1 1 1 −max( ∗ , )−max( , )− ∗ min( , )+min( , )+ − &p,q,r n 2 p 2 q∗ 2 r = n p 2 q 2 r 2 , (5.3) 1 1 1 1 where in (∗) we used the fact that for any p ≥ 1, 1 − max p∗ , 2 = min p , 2 . Combin- ing (5.2) and (5.3), the proof is complete. 5.2. Upper bounds on the operator norm. For tensor products of two spaces, the norm of the identity was estimated in [Sch82].

Proposition 5.2. Let n ∈ N and 1 ≤ p ≤ q ≤ ∞. Then  1 n q q ≤ 2,  1 1 1 min( p + q ,1)− 2 kidk n2 n n p,q n p ≤ 2 ≤ q, `2 →`p ⊗π`q  max 1 + 1 , 1 − 1 n ( p q 2 ) 2 2 ≤ p.

To obtain an upper bound for 3-fold tensor products of `p-spaces, we use the following recursive formula.

Proposition 5.3. Let n ∈ N and 1 ≤ p, q, r ≤ ∞. Then 1 1 min( r , 2 ) kidk n3 n ≤ n · kidk n2 n n . `2 →Xπ `2 →`p ⊗π`q Proof. First, note that

kidk n2 n n = kidk n n ∗ n2 . `2 →`p ⊗π`q (`p ⊗π`q ) →`2

13 n ∗ Let A ∈ (Xπ ) . By the deﬁnition of the injective tensor product norm, we have

2 n ∗ 2 n n n ∗ 2 kidk n n ∗ n · kAk(X ) = kidk n n ∗ n · kAk` →(` ⊗π` ) ≥ kAk n n . (5.4) (`p ⊗π`q ) →`2 π (`p ⊗π`q ) →`2 r p q `r →`2 n Also, for x = (x1, . . . , xn) ∈ `r , we have n " n # X X n2 Ax = Ai,j,kxk ei ⊗ ej ∈ R . i,j=1 k=1 Thus, we obtain

n n 2 2 2 X X kAk 2 = sup kAxk 2 = sup A x `n→`n `n i,j,k k r 2 kxk =1 2 kxk n =1 r `r i,j=1 k=1 Choosing x = n−1/r · ε, ε ∈ {−1, 1}n, we get

n n 2 n 2 2 − 2 X X − 2 X 2 − 1 r r r 3 kAk n n2 ≥ n E Ai,j,kεk = n |Ai,j,k| = n kAk`n . (5.5) `r →`2 2 i,j=1 k=1 i,j,k=1 Combining (5.4) and (5.5), we get

1 r n kidk n n ∗ n2 kAk ≥ kAk n3 . (`p ⊗π`q ) →`2 `2

Equivalently, since kidk n2 n n = kidk n n ∗ n2 , we have `2 →`p ⊗π`q (`p ⊗π`q ) →`2

3 kAk`n 1 2 r ≤ n kidk n2 n n . ` →` ⊗π` kAk 2 p q n ∗ Since A ∈ (Xπ ) was arbitrary, this gives

1 r kidk n3 n ≤ n kidk n2 n n . (5.6) `2 →Xπ `2 →`p ⊗π`q n √ Also, since for all x ∈ , kxk n ≤ kxk n ≤ nkxk n , we have R `∞ `2 `∞

n n 2 n 2 2 X X X 2 1 3 kidk n n ∗ n2 kAk ≥ sup Ai,j,kxk ≥ max |Ai,j,k| ≥ kAk n . (` ⊗π` ) →` `2 p q 2 kxk n =1 1≤k≤n n `r i,j=1 k=1 i,j=1 This implies √ kidk n3 n ≤ n kidk n2 n n . (5.7) `2 →Xπ `2 →`p ⊗π`q Combining (5.6) and (5.7), the result follows. Another useful tool is the following corollary, which follows immediately from Theorem2 in Section3.

1 1 1 1 Corollary 5.4. Let n ∈ N and assume that 1 ≤ p, q, r ≤ ∞ are such that p + q + r ≤ 2 . Then

kidk n3 n 1. `2 →Xπ .

14 1 1 1 1 Proof. Since we assume that p + q + r ≤ 2 , if µ is deﬁned as in (3.4), then we get µ ≤ 2. Hence, we have

kAk n3 ≤ kAk n3 kAk(Xn)∗ . `2 `µ . π Therefore, we have

kidk n3 n = kidk n ∗ n3 1, `2 →Xπ (Xπ ) →`2 . which completes the proof. Using the above upper bounds, we obtain the following.

Lemma 5.5. Let n ∈ N and 1 ≤ p ≤ q ≤ r ≤ ∞. Then we have  1 + 1 n 2 r , r ≤ 2, (5.8a)   min 1 + 1 ,1  ( q r ) n , q ≤ 2 ≤ r, (5.8b)  1 1 1 max( q + r , 2 ) 1 1 1 kidk n3 n p,q,r n , p ≤ 2 ≤ q, + + ≥ 1, (5.8c) `2 →Xπ . p q r  1 + 1 + 1 − 1 n p q r 2 , p ≤ 2 ≤ q, 1 + 1 + 1 ≤ 1, (5.8d)  p q r   max 1 + 1 + 1 , 1 − 1 n ( p q r 2 ) 2 , 2 ≤ p. (5.8e) Proof. The bounds (5.8a), (5.8b), (5.8c) follow from Proposition 5.2 and Proposition 5.3. 1 1 1 1 Next, assume that p, q, r are such that 2 ≤ p + q + r ≤ 1. Then there existsp ˜ ≥ p,q ˜ ≥ q, 1 1 1 1 n r˜ ≥ r such that p˜ + q˜ + r˜ = 2 . Hence, given A ∈ Xπ , we have

1 + 1 + 1 − 1 kAk ≤ n p q r 2 kAk n n n π `p˜⊗π`q˜ ⊗π`r˜ 1 1 1 1 p + q + r − 2 ≤ n kidk n3 n n n kAk n3 `2 →`p˜⊗π`q˜ ⊗π`r˜ `2 1 1 1 1 p + q + r − 2 n kAk n3 , . `2 where in the last inequality we used Corollary 5.4. Hence, we have

1 1 1 1 p + q + r − 2 kidk n3 n n . (5.9) `2 →Xπ . 1 1 1 1 1 1 1 Thus, if we assume that p ≤ 2 ≤ q and p + q + r ≤ 1, then we must have 2 ≤ p + q + r ≤ 1, 1 1 1 1 in which case (5.9) proves (5.8d). Finally, assume that p ≥ 2. If 2 ≤ p + q + r ≤ 1, then (5.9) 1 1 1 1 holds. If p + q + r ≤ 2 , then by Corollary 5.4,

kidk n3 n 1. (5.10) `2 →Xπ . 1 1 1 1 1 1 If we have p + q + r ≥ 1, then since p ≥ 2 we must have q + r ≥ 2 . Hence, using Proposition 5.2 and Proposition 5.3, we have

1 1 1 1 1 1 1 1 1 p +max( q + r , 2 )− 2 p + q + r − 2 kidk n3 n n = n . (5.11) `2 →Xπ . Combining (5.9), (5.10), and (5.11), (5.8e) follows.

15 5.3. Improved lower bounds for some special cases. In this short section we show that the lower bound for the case p ≤ q ≤ 2 ≤ r in TheoremA can be improved for some particular choices of p, q and r. Lemma 5.6. Let n ∈ N and 1 ≤ p ≤ q ≤ 2 ≤ r ≤ ∞. Then we have 1 1 min 1 + 1 ,1 + 1 −1 1 2 + r ( q r ) p q kidk n3 n ≥ max n , n , n . `2 →Xπ Proof. The ﬁrst bound follows from Lemma 5.1. In order to prove the second bound, we n n transfer to a 3-fold tensor that is constructed from a 2-fold tensor B ∈ `q ⊗π `r and the n special vector x = (1, 1,..., 1) ∈ `p . We have

kx ⊗ Bkπ kidk`n3 →Xn ≥ sup 2 π n n 3 B∈`q ⊗π`r kx ⊗ Bk n `2 kBk`n⊗ `n kxk`n = sup q π r p n n 2 n B∈`q ⊗π`r kBk n kxk` `2 2 1 1 p − 2 = n kidk n2 n n `2 →`q ⊗π`r min 1 + 1 ,1 + 1 −1 = n ( q r ) p , where in the last equality we used Proposition 5.2. To obtain the third bound, we again n n n transfer to the 2-fold case, choosing B ∈ `p ⊗π `q and x = (1, 0,..., 0) ∈ `r . Then we have

n n n 1 kx ⊗ Bkπ kBk`p ⊗π`q kxk`r q kidk`n3 →Xn ≥ sup = sup = kidk`n2 →`n⊗ `n = n , 2 π n n 3 n n 2 n 2 p π q B∈`p ⊗π`q kx ⊗ Bk n B∈`p ⊗π`q kBk n kxk` `2 `2 2 where again in the last equality we used Proposition 5.2. This completes the proof. Combining Lemma 5.6 with Lemma 5.5, we obtain the following improvements on the bounds in particular cases. Corollary 5.7. (i) If p = 1 ≤ q ≤ 2 ≤ r, then

1 1 min( q + r ,1) kidk n3 n q,r n . `2 →Xπ In particular, in such case we have

1 1 1 1 n n n min( q − 2 , 2 − r ) vr `p ⊗π `q ⊗π `r q,r n . (ii) If p = q ≤ 2 and r = ∞, then 1 q kidk n3 n p n . `2 →Xπ In particular, in such case we have 1 1 n n n q − 2 vr `p ⊗π `q ⊗π `r p n . (iii) If 1 ≤ p ≤ q = 2 ≤ r, then 1 1 2 + r kidk n3 n p,r n . `2 →Xπ In particular, in such case we have n n n vr `p ⊗π `q ⊗π `r p,r 1.

16 Corollary 5.7 suggests that it is the lower bound that should be improved. See also Corollary 6.1 below for another such indication.

6. The case of k-fold projective tensor products In this section, we brieﬂy present the generalization of our main result to the case of k-fold projective tensor products. Before stating the generalized main result, we ﬁx some notation. Given k ∈ N, 1 ≤ p1 ≤ p2 ≤ · · · ≤ pk ≤ ∞ and n ∈ N, let Xn := `n ⊗ · · · ⊗ `n . (6.1) p p1 π π pk n Also, let j0 be the largest 1 ≤ j ≤ k such that pj ≤ 2 (j0 = 0 if p1 > 2). The space Xp has enough symmetries, and so formula (2.3) gives

k n k/2 1/n vr(X ) n vol k (B n ) kidk n n . (6.2) p n Xp `2 →Xp

In what follows, &p and .p mean inequalities with an implied constant that depends only on p1, . . . , pk. All the tools that were used above can be generalized to the case of k-fold tensor products by straightforward induction.

n Regarding the volume of BXp , note that in the k-fold version of Chevet’s inequality (see Lemma 6.2 below for a complete statement), there is an additional k factor since the upper bound is now comprised of k terms, and also note that the lower bound of the volume of n − k−j0 B now contains a factor of 2 2 , since for each j with p ≥ 2, the use of Khinchine’s Xp √ j inequality incurs a factor of 1/ 2. As a result, we have in the general case, Pk−1 1 1 1 Pk−1 1 1 1 k−j0 − min , − −1 − min , − −1 − j=1 pj 2 pk j=1 pj 2 pk 2 k n 2 n .p voln (BXp ) .p k n . Regarding the norm of the identity, an analogue of TheoremC follows by simply using induction. The only exception is Theorem2, which in the general case still gives Pk 1 1 kidk nk n 1 whenever j=1 ≤ . `2 →Xp . pj 2 Combining those tools, TheoremA can be generalized in the following way:

Theorem D. Let k ≥ 3. Let 1 ≤ p1 ≤ p2 ≤ · · · ≤ pk ≤ ∞ and n ∈ N. Then the following estimates hold:

(1) If pk ≤ 2, then n 1 .p vr(Xπ ) .p k. Pk 1 (2) If ≥ 1, pk−1 ≤ 2 < pk, then j=1 pj min 1 + 1 ,1 − 1 − 1 n pk−1 pk pk 2 1 .p vr(Xπ ) .p k n . Pk 1 (3) If ≥ 1, pk−1 > 2, then j=1 pj max 1 + 1 , 1 − 1 − 1 n pk−1 pk 2 pk−1 pk 1 .p vr(Xπ ) .p k n (4) If Pk 1 ≥ 1, and If Pk 1 ≤ 1 , then j=1 pj j=2 pj 2

k 1 Pk 1 1 Pk 1 − j=2 − j=2 − 2 2 pj n 2 pj 2 n .p vr(Xπ ) .p k n

17 (5) If 1 ≤ Pk 1 ≤ 1, then 2 j=1 pj k max 1 − 1 ,0 max 1 − 1 ,0 − 2 p 2 n p 2 2 n 1 .p vr(Xπ ) .p k n 1 . (6) If Pk 1 ≤ 1 , then j=1 pj 2

k 1 Pk 1 1 Pk 1 − j=1 − j=1 − 2 2 pj n 2 pj 2 n .p vr(Xπ ) .p k n . n Considering copies of the same `p space, it was shown in [DP09], that if one considers the space k n n n n ⊗π`p := `p ⊗π `p ⊗π · · · ⊗π `p , | {z } k times then the following holds true: ( k n 1 p ≤ 2k, vr ⊗π`p p 1 − k (6.3) n 2 p p ≥ 2k. Using TheoremD, we can get a result in the spirit of (6.3), with worse dependence on k, and in some case, with worse dependence on n as well: Corollary 6.1. The following holds: k n (1) If p ≤ 4 or k ≤ p ≤ 2k then 1 .p vr ⊗π`p .p k. 1 2 k n 2 − p (2) If 4 ≤ p ≤ k then 1 .p vr ⊗π`p .p k n . k 1 k 1 k − 2 − p k n 2 − p (3) If p ≥ 2k then 2 2 n .p vr ⊗π`p .p k n . Corollary 6.1 combined with (6.3) suggest that it is the lower bound that should be improved in TheoremD. Finally, for the sake of completeness, let us state the k-fold version of Chevet’s inequality that plays a major role in the proof of TheoremD: Lemma 6.2. Let k ∈ and {x }nj ⊆ X , 1 ≤ j ≤ k be sequences in the Banach spaces N ij ij =1 j

(X1, k · kX1 ),..., (Xk, k · kXk ), respectively. Then

k " # n X X Y X g x ⊗ · · · ⊗ x ≤ {x } g x . i1,...,ik i1 ik ij ij ω,2 E ij ij 0 1≤im≤n j=1 j 6=j ij =1 Xj 1≤m≤k X1⊗···⊗Xk

7. Remarks on the cotype of projective tensor products As already mentioned in the introduction, from our main result and its k-fold generalization, we obtain information on the cotype of the 3-fold and k-fold projective tensor products. Consider now the inﬁnite dimensional space

X := `p ⊗π `q ⊗π `r, n n n where 1 ≤ p ≤ q ≤ r ≤ ∞. Since the space Xπ = `p ⊗π `q ⊗π `r is a ﬁnite dimensional subspace of X, it follows by Theorem1 that if n n lim vr `p ⊗π `q ⊗π `r = ∞, n→∞

18 then X does not have cotype 2, that is C2(X) = ∞. In order to obtain a result about cotype α > 2, recall the following result from [TJ79]: if (X, k·kX ) is n-dimensional, then there exist x1, . . . , xn ∈ X such that n n 1/2 X 2 X kxikX & C2(X) E εixi . X i=1 i=1 n n n n n In particular, in the space Xπ = `p ⊗π `q ⊗π `r , there exist x1, . . . , xn3 ∈ Xπ such that

n3 n3 1/2 n X X 2 C2(Xπ ) E εixi . kxikπ π i=1 i=1 n3 n3 (∗) 1/α 3 1 − 1 X α 3 1 − 1 n X ( 2 α ) ( 2 α ) ≤ n kxikπ ≤ n Cα(Xπ ) E εixi , π i=1 i=1

1 1 n 3( α − 2 ) n where in (∗) we used H¨older’sinequality. Altogether, we have Cα(Xπ ) & n C2(Xπ ). This fact, together with TheoremA, gives the following result.

Corollary 7.1. Let X = `p ⊗π `q ⊗π `r with 1 ≤ p ≤ q ≤ r ≤ ∞. Then Cα(X) = ∞ in the following cases: 1 1 1 1 1 1 3 • p ≤ 2, q + r < 2 , and p + q + r ≥ 1, and for α < 1 1 ; 1+ q + r 1 1 1 3 • p < 2, p + q + r ≤ 1, and for α < 1 ; 2− p 1 1 1 1 3 • p + q + r < 2 , and for α < 1 1 1 . 1+ p + q + r Finally, let us comment on the case of k-fold tensor products. Let

X := `p1 ⊗π · · · ⊗π `pk , n where 1 ≤ p1 ≤ · · · ≤ pk ≤ ∞. If Xp is deﬁned as in (6.1), then a similar argument as 1 1 n k( α − 2 ) n above implies that Cα(Xp ) & n C2(Xp ). Thus, using TheoremD, a similar result to Corollary 7.1 can be obtained. The result then reads as follows.

Corollary 7.2. Let X = `p1 ⊗π · · · ⊗π `pk with 1 ≤ p1 ≤ · · · ≤ pk ≤ ∞. Then Cα(X) = ∞ in the following cases: Pk 1 Pk 1 1 k • ≥ 1 and < and for α < k−1 k 1 ; j=1 pj j=2 pj 2 +P 2 j=2 pj Pk 1 k • ≤ 1 and p1 < 2 and for α < k+1 1 ; j=1 pj − 2 p1 Pk 1 1 k • < and for α < k−1 n 1 . j=1 pj 2 +P 2 j=1 pj References [AS06] G. Aubrun and S. Szarek, Tensor products of convex sets and the volume of separable states on N qudits, Phys. Rev. A. 73 (2006), no. 022109. [ASW11] G. Aubrun, S. Szarek, and E. Werner, Hastings’s Additivity Counterexample via Dvoretzky’s The- orem, Commun. Math. Phys. 305 (2011), no. 1, 85–97. [ASW10] , Nonadditivity of R´enyientropy and Dvoretzky’s theorem, J. Math. Phys. 51 (2010), no. 2. [AF96] A. Arias and J. D. Farmer, On the structure of tensor products of lp-spaces, Paciﬁc J. Math. 175 (1996), no. 1, 13–37.

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School of Mathematical & Physical Sciences, University of Newcastle, Callaghan, NSW 2308, Australia E-mail address: [email protected]

School of Mathematics & Physical Sciences, University of Hull, Cottingham Road, Hull, HU6 7RX, United Kingdom E-mail address: [email protected]

Mathematisches Seminar, Christian-Albrechts-University Kiel, Ludewig-Meyn-Straße 4, 24098 Kiel, Germany E-mail address: [email protected]

Department of Mathematical and Statistical Sciences, University of Alberta, 632 Cen- tral Academic Building, Edmonton, AB T6G 2G1, Canada E-mail address: [email protected]

Department of Mathematics, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, OH 44106, USA E-mail address: [email protected]